The generalized lower triangular process prior

2026-10-07

AI Summary

This study addresses the poor interpretability of multivariate factor models caused by rotational invariance, as well as the complex Bayesian implementation and challenging prior specification associated with generalized lower triangular structures. To overcome these limitations, this work proposes a generalized lower triangular process prior that supports dual sparsity at both global and within-component levels. The proposed approach substantially simplifies prior hyperparameter specification and establishes a theoretical connection to the Indian Buffet Process. Furthermore, an efficient Gibbs sampler incorporating Metropolis-Hastings steps is designed for posterior inference. Simulation studies and an empirical analysis of Big Five personality data demonstrate that the method automatically infers the number of factors while discovering sparse structures, thereby significantly enhancing model interpretability and practical utility.

Abstract

In the analysis of multivariate data, factor models represent a powerful technique for both reducing dimensionality and facilitating qualitative understanding of the latent determinants governing the observed data. Interpretability, however, is often hindered by the non-identifiability of factor loading matrices due to rotational invariance. A prevailing strategy to address this issue is to consider positive lower triangular structures. A relaxation of the latter condition has been recently proposed in the literature with the so called generalized lower triangular structure. However, its Bayesian implementation presents substantial challenges, requiring complex reversible-jump algorithms and making prior elicitation difficult because of the limited interpretability of the model parameters. In this paper, we introduce a generalized lower triangular process prior that allows both global and within-component sparsity structures for learning both the number of factors and possible sparsity structures within the vector of observations. The proposed approach provides straightforward prior parameters elicitation exploiting possibly different prior information on the rank and sparsity characteristics. We also explore connections with the Indian buffet process, providing further insight into the sparsity structure induced by the proposed prior. Posterior computation can be performed resorting to a Gibbs sampler with Metropolis-Hastings moves. The efficacy of the proposed method is demonstrated through comprehensive simulations and the analysis of the Big Five Personality Test data.

Problem

• factor models
• identifiability
• generalized lower triangular structure
• Bayesian implementation
• sparsity

Innovation

• Generalized lower triangular process prior
• Factor models
• Sparsity structure
• Bayesian inference
• Indian buffet process